/*=========================================================================
 *
 *  Copyright UMC Utrecht and contributors
 *
 *  Licensed under the Apache License, Version 2.0 (the "License");
 *  you may not use this file except in compliance with the License.
 *  You may obtain a copy of the License at
 *
 *        http://www.apache.org/licenses/LICENSE-2.0.txt
 *
 *  Unless required by applicable law or agreed to in writing, software
 *  distributed under the License is distributed on an "AS IS" BASIS,
 *  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 *  See the License for the specific language governing permissions and
 *  limitations under the License.
 *
 *=========================================================================*/

#include "itkCMAEvolutionStrategyOptimizer.h"
#include "itkSymmetricEigenAnalysis.h"
#include <vnl/vnl_math.h>
#include <algorithm>
#include <cmath>
#include "itkCommand.h"
#include "itkEventObject.h"
#include "itkMacro.h"

namespace itk
{

/**
 * ******************** Constructor *************************
 */

CMAEvolutionStrategyOptimizer::CMAEvolutionStrategyOptimizer()
{
  itkDebugMacro("Constructor");

} // end constructor


/**
 * ******************* PrintSelf *********************
 */

void
CMAEvolutionStrategyOptimizer::PrintSelf(std::ostream & os, Indent indent) const
{
  Superclass::PrintSelf(os, indent);

  // os << indent << "m_RandomGenerator: " << this->m_RandomGenerator << std::endl;
  os << indent << "m_CurrentValue: " << this->m_CurrentValue << std::endl;
  os << indent << "m_CurrentIteration: " << this->m_CurrentIteration << std::endl;
  os << indent << "m_StopCondition: " << this->m_StopCondition << std::endl;
  os << indent << "m_Stop: " << this->m_Stop << std::endl;

  os << indent << "m_UseCovarianceMatrixAdaptation: " << this->m_UseCovarianceMatrixAdaptation << std::endl;
  os << indent << "m_PopulationSize: " << this->m_PopulationSize << std::endl;
  os << indent << "m_NumberOfParents: " << this->m_NumberOfParents << std::endl;
  os << indent << "m_UpdateBDPeriod: " << this->m_UpdateBDPeriod << std::endl;

  os << indent << "m_EffectiveMu: " << this->m_EffectiveMu << std::endl;
  os << indent << "m_ConjugateEvolutionPathConstant: " << this->m_ConjugateEvolutionPathConstant << std::endl;
  os << indent << "m_SigmaDampingConstant: " << this->m_SigmaDampingConstant << std::endl;
  os << indent << "m_CovarianceMatrixAdaptationConstant: " << this->m_CovarianceMatrixAdaptationConstant << std::endl;
  os << indent << "m_EvolutionPathConstant: " << this->m_EvolutionPathConstant << std::endl;
  os << indent << "m_CovarianceMatrixAdaptationWeight: " << this->m_CovarianceMatrixAdaptationWeight << std::endl;
  os << indent << "m_ExpectationNormNormalDistribution: " << this->m_ExpectationNormNormalDistribution << std::endl;
  os << indent << "m_HistoryLength: " << this->m_HistoryLength << std::endl;

  os << indent << "m_CurrentSigma: " << this->m_CurrentSigma << std::endl;
  os << indent << "m_Heaviside: " << this->m_Heaviside << std::endl;
  os << indent << "m_CurrentMaximumD: " << this->m_CurrentMaximumD << std::endl;
  os << indent << "m_CurrentMinimumD: " << this->m_CurrentMinimumD << std::endl;

  os << indent << "m_MaximumNumberOfIterations: " << this->m_MaximumNumberOfIterations << std::endl;
  os << indent << "m_UseDecayingSigma: " << this->m_UseDecayingSigma << std::endl;
  os << indent << "m_InitialSigma: " << this->m_InitialSigma << std::endl;
  os << indent << "m_SigmaDecayA: " << this->m_SigmaDecayA << std::endl;
  os << indent << "m_SigmaDecayAlpha: " << this->m_SigmaDecayAlpha << std::endl;
  os << indent << "m_RecombinationWeightsPreset: " << this->m_RecombinationWeightsPreset << std::endl;
  os << indent << "m_MaximumDeviation: " << this->m_MaximumDeviation << std::endl;
  os << indent << "m_MinimumDeviation: " << this->m_MinimumDeviation << std::endl;
  os << indent << "m_PositionToleranceMin: " << this->m_PositionToleranceMin << std::endl;
  os << indent << "m_PositionToleranceMax: " << this->m_PositionToleranceMax << std::endl;
  os << indent << "m_ValueTolerance: " << this->m_ValueTolerance << std::endl;

  os << indent << "m_RecombinationWeights: " << this->m_RecombinationWeights << std::endl;
  os << indent << "m_C: " << this->m_C << std::endl;
  os << indent << "m_B: " << this->m_B << std::endl;
  os << indent << "m_D: " << this->m_D.diagonal() << std::endl;

  // template:
  // os << indent << ": " << this-> << std::endl;

} // end PrintSelf;


/**
 * ******************* StartOptimization *********************
 */

void
CMAEvolutionStrategyOptimizer::StartOptimization()
{
  itkDebugMacro("StartOptimization");

  /** Reset some variables */
  this->m_CurrentValue = NumericTraits<MeasureType>::Zero;
  this->m_CurrentIteration = 0;
  this->m_Stop = false;
  this->m_StopCondition = Unknown;

  /** Get the number of parameters; checks also if a cost function has been set at all.
   * if not: an exception is thrown */
  this->GetScaledCostFunction()->GetNumberOfParameters();

  /** Initialize the scaledCostFunction with the currently set scales */
  this->InitializeScales();

  /** Set the current position as the scaled initial position */
  this->SetCurrentPosition(this->GetInitialPosition());

  /** Compute default values for a lot of constants */
  this->InitializeConstants();

  /** Resize/Initialize variables used that are updated during optimisation */
  this->InitializeProgressVariables();

  /** Resize/Initialize B, C, and D */
  this->InitializeBCD();

  if (!this->m_Stop)
  {
    this->ResumeOptimization();
  }

} // end StartOptimization


/**
 * ******************* ResumeOptimization *********************
 */

void
CMAEvolutionStrategyOptimizer::ResumeOptimization()
{
  itkDebugMacro("ResumeOptimization");

  this->m_StopCondition = Unknown;

  this->InvokeEvent(StartEvent());

  try
  {
    this->m_CurrentValue = this->GetScaledValue(this->GetScaledCurrentPosition());
  }
  catch (ExceptionObject & err)
  {
    this->m_StopCondition = MetricError;
    this->StopOptimization();
    throw err;
  }

  /** Test if not by chance we are already converged */
  bool convergence = this->TestConvergence(true);
  if (convergence)
  {
    this->StopOptimization();
  }

  /** Start iterating */
  while (!this->m_Stop)
  {
    this->GenerateOffspring();
    this->SortCostFunctionValues();

    /** Something may have gone wrong during evaluation of the cost function values */
    if (this->m_Stop)
    {
      break;
    }

    this->AdvanceOneStep();

    /** Something may have gone wrong during evalution of the current value */
    if (this->m_Stop)
    {
      break;
    }

    /** Give the user opportunity to observe progress (current value/position/sigma etc.) */
    this->InvokeEvent(IterationEvent());

    if (this->m_Stop)
    {
      break;
    }

    /** Prepare for next iteration */
    this->UpdateConjugateEvolutionPath();
    this->UpdateHeaviside();
    this->UpdateEvolutionPath();
    this->UpdateC();
    this->UpdateSigma();
    this->UpdateBD();
    this->FixNumericalErrors();

    /** Test if convergence has occured in some sense */
    convergence = this->TestConvergence(false);
    if (convergence)
    {
      this->StopOptimization();
      break;
    }

    /** Next iteration */
    ++(this->m_CurrentIteration);

  } // end while !m_Stop

} // end ResumeOptimization


/**
 * *********************** StopOptimization *****************************
 */

void
CMAEvolutionStrategyOptimizer::StopOptimization()
{
  itkDebugMacro("StopOptimization");
  this->m_Stop = true;
  this->InvokeEvent(EndEvent());
} // end StopOptimization()


/**
 * ****************** InitializeConstants *********************
 */

void
CMAEvolutionStrategyOptimizer::InitializeConstants(void)
{
  itkDebugMacro("InitializeConstants");

  /** Get the number of parameters from the cost function */
  const unsigned int numberOfParameters = this->GetScaledCostFunction()->GetNumberOfParameters();

  /** m_PopulationSize (if not provided by the user) */
  if (this->m_PopulationSize == 0)
  {
    this->m_PopulationSize =
      4 + static_cast<unsigned int>(std::floor(3.0 * std::log(static_cast<double>(numberOfParameters))));
  }

  /** m_NumberOfParents (if not provided by the user) */
  if (this->m_NumberOfParents == 0)
  {
    this->m_NumberOfParents = this->m_PopulationSize / 2;
  }

  /** Some casts/aliases: */
  const unsigned int N = numberOfParameters;
  const double       Nd = static_cast<double>(N);
  const unsigned int lambda = this->m_PopulationSize;
  const double       lambdad = static_cast<double>(lambda);
  const unsigned int mu = this->m_NumberOfParents;
  const double       mud = static_cast<double>(mu);

  /** m_RecombinationWeights */
  this->m_RecombinationWeights.SetSize(mu);
  this->m_RecombinationWeights.Fill(1.0); // "equal" preset
  if (this->m_RecombinationWeightsPreset == "linear")
  {
    for (unsigned int i = 0; i < mu; ++i)
    {
      this->m_RecombinationWeights[i] = mud + 1.0 - static_cast<double>(i + 1);
    }
  }
  else if (this->m_RecombinationWeightsPreset == "superlinear")
  {
    const double logmud = std::log(mud + 1.0);
    for (unsigned int i = 0; i < mu; ++i)
    {
      this->m_RecombinationWeights[i] = logmud - std::log(static_cast<double>(i + 1));
    }
  }
  this->m_RecombinationWeights /= this->m_RecombinationWeights.sum();

  /** m_EffectiveMu */
  this->m_EffectiveMu = 1.0 / this->m_RecombinationWeights.squared_magnitude();
  if (this->m_EffectiveMu >= lambdad)
  {
    itkExceptionMacro(<< "The RecombinationWeights have unreasonable values!");
  }
  /** alias: */
  const double mueff = this->m_EffectiveMu;

  /** m_ConjugateEvolutionPathConstant (c_\sigma) */
  this->m_ConjugateEvolutionPathConstant = (mueff + 2.0) / (Nd + mueff + 3.0);

  /** m_SigmaDampingConstant */
  this->m_SigmaDampingConstant = this->m_ConjugateEvolutionPathConstant +
                                 (1.0 + 2.0 * std::max(0.0, std::sqrt((mueff - 1.0) / (Nd + 1.0)) - 1.0)) *
                                   std::max(0.3, 1.0 - Nd / static_cast<double>(this->m_MaximumNumberOfIterations));

  /** m_CovarianceMatrixAdaptationWeight (\mu_cov)*/
  this->m_CovarianceMatrixAdaptationWeight = mueff;
  /** alias: */
  const double mucov = this->m_CovarianceMatrixAdaptationWeight;

  /** m_CovarianceMatrixAdaptationConstant (c_cov) */
  this->m_CovarianceMatrixAdaptationConstant =
    (1.0 / mucov) * 2.0 / vnl_math::sqr(Nd + std::sqrt(2.0)) +
    (1.0 - 1.0 / mucov) * std::min(1.0, (2.0 * mueff - 1.0) / (vnl_math::sqr(Nd + 2.0) + mueff));
  /** alias: */
  const double c_cov = this->m_CovarianceMatrixAdaptationConstant;

  /** Update only every 'period' iterations */
  if (this->m_UpdateBDPeriod == 0)
  {
    this->m_UpdateBDPeriod = static_cast<unsigned int>(std::floor(1.0 / c_cov / Nd / 10.0));
  }
  this->m_UpdateBDPeriod = std::max(static_cast<unsigned int>(1), this->m_UpdateBDPeriod);
  if (this->m_UpdateBDPeriod >= this->m_MaximumNumberOfIterations)
  {
    this->SetUseCovarianceMatrixAdaptation(false);
  }

  /** m_EvolutionPathConstant (c_c)*/
  this->m_EvolutionPathConstant = 4.0 / (Nd + 4.0);

  /** m_ExpectationNormNormalDistribution */
  this->m_ExpectationNormNormalDistribution =
    std::sqrt(Nd) * (1.0 - 1.0 / (4.0 * Nd) + 1.0 / (21.0 * vnl_math::sqr(Nd)));

  /** m_HistoryLength */
  this->m_HistoryLength = static_cast<unsigned long>(std::min(
    this->GetMaximumNumberOfIterations(), 10 + static_cast<unsigned long>(std::ceil(3.0 * 10.0 * Nd / lambdad))));

} // end InitializeConstants


/**
 * ****************** InitializeProgressVariables *********************
 */

void
CMAEvolutionStrategyOptimizer::InitializeProgressVariables(void)
{
  itkDebugMacro("InitializeProgressVariables");

  /** Get the number of parameters from the cost function */
  const unsigned int numberOfParameters = this->GetScaledCostFunction()->GetNumberOfParameters();

  /** Some casts/aliases: */
  const unsigned int N = numberOfParameters;
  const unsigned int lambda = this->m_PopulationSize;

  /** CurrentSigma */
  this->m_CurrentSigma = this->GetInitialSigma();

  /** Heaviside */
  this->m_Heaviside = 0.0;

  /** m_SearchDirs */
  const ParametersType zeroParam(N, 0.0);
  this->m_SearchDirs.clear();
  this->m_SearchDirs.resize(lambda, zeroParam);

  /** m_NormalizedSearchDirs */
  this->m_NormalizedSearchDirs.clear();
  this->m_NormalizedSearchDirs.resize(lambda, zeroParam);

  /** m_CostFunctionValues */
  this->m_CostFunctionValues.clear();

  /** m_CurrentScaledStep */
  this->m_CurrentScaledStep.SetSize(N);
  this->m_CurrentScaledStep.Fill(0.0);

  /** m_CurrentNormalizedStep */
  this->m_CurrentNormalizedStep.SetSize(N);
  this->m_CurrentNormalizedStep.Fill(0.0);

  /** m_EvolutionPath */
  this->m_EvolutionPath.SetSize(N);
  this->m_EvolutionPath.Fill(0.0);

  /** m_ConjugateEvolutionPath */
  this->m_ConjugateEvolutionPath.SetSize(N);
  this->m_ConjugateEvolutionPath.Fill(0.0);

  /** m_MeasureHistory */
  this->m_MeasureHistory.clear();

  /** Maximum and minimum square root eigenvalues */
  this->m_CurrentMaximumD = 1.0;
  this->m_CurrentMinimumD = 1.0;

} // end InitializeProgressVariables


/**
 * ****************** InitializeBCD *********************
 */

void
CMAEvolutionStrategyOptimizer::InitializeBCD(void)
{
  itkDebugMacro("InitializeBCD");

  if (this->GetUseCovarianceMatrixAdaptation())
  {
    /** Get the number of parameters from the cost function */
    const unsigned int numberOfParameters = this->GetScaledCostFunction()->GetNumberOfParameters();

    /** Some casts/aliases: */
    const unsigned int N = numberOfParameters;

    /** Resize */
    this->m_B.SetSize(N, N);
    this->m_C.SetSize(N, N);
    this->m_D.set_size(N);

    /** Initialize */
    this->m_B.Fill(0.0);
    this->m_C.Fill(0.0);
    this->m_B.fill_diagonal(1.0);
    this->m_C.fill_diagonal(1.0);
    this->m_D.fill(1.0);
  }
  else
  {
    /** Clear */
    this->m_B.SetSize(0, 0);
    this->m_C.SetSize(0, 0);
    this->m_D.clear();
  }

} // end InitializeBCD


/**
 * ****************** GenerateOffspring *********************
 */

void
CMAEvolutionStrategyOptimizer::GenerateOffspring(void)
{
  itkDebugMacro("GenerateOffspring");

  /** Get the number of parameters from the cost function */
  const unsigned int numberOfParameters = this->GetScaledCostFunction()->GetNumberOfParameters();

  /** Some casts/aliases: */
  const unsigned int N = numberOfParameters;
  const unsigned int lambda = this->m_PopulationSize;

  /** Clear the old values */
  this->m_CostFunctionValues.clear();

  /** Fill the m_NormalizedSearchDirs and SearchDirs */
  unsigned int lam = 0;
  unsigned int nrOfFails = 0;
  while (lam < lambda)
  {
    /** draw from distribution N(0,I) */
    for (unsigned int par = 0; par < N; ++par)
    {
      this->m_NormalizedSearchDirs[lam][par] = this->m_RandomGenerator->GetNormalVariate();
    }
    /** Make like it was drawn from N(0,C) */
    if (this->GetUseCovarianceMatrixAdaptation())
    {
      this->m_SearchDirs[lam] = this->m_B * (this->m_D * this->m_NormalizedSearchDirs[lam]);
    }
    else
    {
      this->m_SearchDirs[lam] = this->m_NormalizedSearchDirs[lam];
    }
    /** Make like it was drawn from N( 0, sigma^2 C ) */
    this->m_SearchDirs[lam] *= this->m_CurrentSigma;

    /** Compute the cost function */
    MeasureType costFunctionValue = 0.0;
    /** x_lam = m + d_lam */
    ParametersType x_lam = this->GetScaledCurrentPosition();
    x_lam += this->m_SearchDirs[lam];
    try
    {
      costFunctionValue = this->GetScaledValue(x_lam);
    }
    catch (ExceptionObject & err)
    {
      ++nrOfFails;
      /** try another parameter vector if we haven't tried that for 10 times already */
      if (nrOfFails <= 10)
      {
        continue;
      }
      else
      {
        this->m_StopCondition = MetricError;
        this->StopOptimization();
        throw err;
      }
    }
    /** Successfull cost function evaluation */
    this->m_CostFunctionValues.push_back(MeasureIndexPairType(costFunctionValue, lam));

    /** Reset the number of failed cost function evaluations */
    nrOfFails = 0;

    /** next offspring member */
    ++lam;
  }

} // end GenerateOffspring


/**
 * ****************** SortCostFunctionValues *********************
 */

void
CMAEvolutionStrategyOptimizer::SortCostFunctionValues(void)
{
  itkDebugMacro("SortCostFunctionValues");

  /** Sort the cost function values in order of increasing cost function value */
  std::sort(this->m_CostFunctionValues.begin(), this->m_CostFunctionValues.end());

  /** Store the best value in the history, and remove the oldest entry of the
   * the history if the history exceeds the HistoryLength */
  this->m_MeasureHistory.push_front(this->m_CostFunctionValues[0].first);
  if (this->m_MeasureHistory.size() > this->m_HistoryLength)
  {
    this->m_MeasureHistory.pop_back();
  }

} // end SortCostFunctionValues


/**
 * ****************** AdvanceOneStep *********************
 */

void
CMAEvolutionStrategyOptimizer::AdvanceOneStep(void)
{
  itkDebugMacro("AdvanceOneStep");

  /** Some casts/aliases: */
  const unsigned int mu = this->m_NumberOfParents;

  /** Compute the CurrentScaledStep, using the RecombinationWeights and
   * the sorted CostFunctionValues-vector.
   * On the fly, also compute the CurrentNormalizedStep */
  this->m_CurrentScaledStep.Fill(0.0);
  this->m_CurrentNormalizedStep.Fill(0.0);
  for (unsigned int m = 0; m < mu; ++m)
  {
    const unsigned int lam = this->m_CostFunctionValues[m].second;
    const double       weight = this->m_RecombinationWeights[m];
    this->m_CurrentScaledStep += (weight * this->m_SearchDirs[lam]);
    this->m_CurrentNormalizedStep += (weight * this->m_NormalizedSearchDirs[lam]);
  }

  /** Set the new current position */
  ParametersType newPos = this->GetScaledCurrentPosition();
  newPos += this->GetCurrentScaledStep();
  this->SetScaledCurrentPosition(newPos);

  /** Compute the cost function at the new position */
  try
  {
    this->m_CurrentValue = this->GetScaledValue(this->GetScaledCurrentPosition());
  }
  catch (ExceptionObject & err)
  {
    this->m_StopCondition = MetricError;
    this->StopOptimization();
    throw err;
  }
} // end AdvanceOneStep


/**
 * ****************** UpdateConjugateEvolutionPath *********************
 */

void
CMAEvolutionStrategyOptimizer::UpdateConjugateEvolutionPath(void)
{
  itkDebugMacro("UpdateConjugateEvolutionPath");

  /** Some casts/aliases: */
  const double c_sigma = this->m_ConjugateEvolutionPathConstant;

  /** Update p_sigma */
  const double factor = std::sqrt(c_sigma * (2.0 - c_sigma) * this->m_EffectiveMu);
  this->m_ConjugateEvolutionPath *= (1.0 - c_sigma);
  if (this->GetUseCovarianceMatrixAdaptation())
  {
    this->m_ConjugateEvolutionPath += (factor * (this->m_B * this->m_CurrentNormalizedStep));
  }
  else
  {
    this->m_ConjugateEvolutionPath += (factor * this->m_CurrentNormalizedStep);
  }

} // end UpdateConjugateEvolutionPath


/**
 * ****************** UpdateHeaviside *********************
 */

void
CMAEvolutionStrategyOptimizer::UpdateHeaviside(void)
{
  itkDebugMacro("UpdateHeaviside");

  /** Get the number of parameters from the cost function */
  const unsigned int numberOfParameters = this->GetScaledCostFunction()->GetNumberOfParameters();

  /** Some casts/aliases: */
  const unsigned int N = numberOfParameters;
  const double       Nd = static_cast<double>(N);
  const double       c_sigma = this->m_ConjugateEvolutionPathConstant;
  const int          nextit = static_cast<int>(this->GetCurrentIteration() + 1);
  const double       chiN = this->m_ExpectationNormNormalDistribution;

  /** Compute the Heaviside function: */
  this->m_Heaviside = false;
  const double normps = this->m_ConjugateEvolutionPath.magnitude();
  const double denom = std::sqrt(1.0 - std::pow(1.0 - c_sigma, 2 * nextit));
  const double righthandside = 1.5 + 1.0 / (Nd - 0.5);
  if ((normps / denom / chiN) < righthandside)
  {
    this->m_Heaviside = true;
  }

} // end UpdateHeaviside


/**
 * ****************** UpdateEvolutionPath *********************
 */

void
CMAEvolutionStrategyOptimizer::UpdateEvolutionPath(void)
{
  itkDebugMacro("UpdateEvolutionPath");

  /** Some casts/aliases: */
  const double c_c = this->m_EvolutionPathConstant;

  /** Compute the evolution path p_c */
  this->m_EvolutionPath *= (1.0 - c_c);
  if (this->m_Heaviside)
  {
    const double factor = std::sqrt(c_c * (2.0 - c_c) * this->m_EffectiveMu) / this->m_CurrentSigma;
    this->m_EvolutionPath += (factor * this->m_CurrentScaledStep);
  }

} // end UpdateEvolutionPath


/**
 * ****************** UpdateC *********************
 */

void
CMAEvolutionStrategyOptimizer::UpdateC(void)
{
  itkDebugMacro("UpdateC");

  if (!(this->GetUseCovarianceMatrixAdaptation()))
  {
    /** We don't need C */
    return;
  }

  /** Get the number of parameters from the cost function */
  const unsigned int numberOfParameters = this->GetScaledCostFunction()->GetNumberOfParameters();

  /** Some casts/aliases: */
  const unsigned int N = numberOfParameters;
  const unsigned int mu = this->m_NumberOfParents;
  const double       c_c = this->m_EvolutionPathConstant;
  const double       c_cov = this->m_CovarianceMatrixAdaptationConstant;
  const double       mu_cov = this->m_CovarianceMatrixAdaptationWeight;
  const double       sigma = this->m_CurrentSigma;

  /** Multiply old m_C with some factor */
  double oldCfactor = 1.0 - c_cov;
  if (!this->m_Heaviside)
  {
    oldCfactor += (c_cov * c_c * (2.0 - c_c) / mu_cov);
  }
  this->m_C *= oldCfactor;

  /** Do rank-one update */
  const double rankonefactor = c_cov / mu_cov;
  for (unsigned int i = 0; i < N; ++i)
  {
    const double evolutionPath_i = this->m_EvolutionPath[i];
    for (unsigned int j = 0; j < N; ++j)
    {
      const double update = rankonefactor * evolutionPath_i * this->m_EvolutionPath[j];
      this->m_C[i][j] += update;
    }
  }

  /** Do rank-mu update */
  const double rankmufactor = c_cov * (1.0 - 1.0 / mu_cov);
  for (unsigned int m = 0; m < mu; ++m)
  {
    const unsigned int lam = this->m_CostFunctionValues[m].second;
    const double       sqrtweight = std::sqrt(this->m_RecombinationWeights[m]);
    ParametersType     weightedSearchDir = this->m_SearchDirs[lam];
    weightedSearchDir *= (sqrtweight / sigma);
    for (unsigned int i = 0; i < N; ++i)
    {
      const double weightedSearchDir_i = weightedSearchDir[i];
      for (unsigned int j = 0; j < N; ++j)
      {
        const double update = rankmufactor * weightedSearchDir_i * weightedSearchDir[j];
        this->m_C[i][j] += update;
      }
    }
  } // end for m

} // end UpdateC


/**
 * ****************** UpdateSigma *********************
 */

void
CMAEvolutionStrategyOptimizer::UpdateSigma(void)
{
  itkDebugMacro("UpdateSigma");

  if (this->GetUseDecayingSigma())
  {
    const double it = static_cast<double>(this->GetCurrentIteration());
    const double num = std::pow(this->m_SigmaDecayA + it, this->m_SigmaDecayAlpha);
    const double den = std::pow(this->m_SigmaDecayA + it + 1.0, this->m_SigmaDecayAlpha);
    this->m_CurrentSigma *= num / den;
  }
  else
  {
    const double normps = this->m_ConjugateEvolutionPath.magnitude();
    const double chiN = this->m_ExpectationNormNormalDistribution;
    const double c_sigma = this->m_ConjugateEvolutionPathConstant;
    const double d_sigma = this->m_SigmaDampingConstant;
    this->m_CurrentSigma *= std::exp((normps / chiN - 1.0) * c_sigma / d_sigma);
  }

} // end UpdateSigma


/**
 * ****************** UpdateBD *********************
 */

void
CMAEvolutionStrategyOptimizer::UpdateBD(void)
{
  itkDebugMacro("UpdateBD");

  /** Get the number of parameters from the cost function */
  const unsigned int numberOfParameters = this->GetScaledCostFunction()->GetNumberOfParameters();

  /** Some casts/aliases: */
  const unsigned int N = numberOfParameters;
  const int          nextit = static_cast<int>(this->GetCurrentIteration() + 1);

  /** Update only every 'm_UpdateBDPeriod' iterations */
  unsigned int periodover = nextit % this->m_UpdateBDPeriod;

  if (!(this->GetUseCovarianceMatrixAdaptation()) || (periodover != 0))
  {
    /** We don't need to update B and D */
    return;
  }

  typedef itk::SymmetricEigenAnalysis<CovarianceMatrixType, EigenValueMatrixType, CovarianceMatrixType>
    EigenAnalysisType;

  /** In the itkEigenAnalysis only the upper triangle of the matrix will be accessed, so
   * we do not need to make sure the matrix is symmetric, like in the
   * matlab code. Just run the eigenAnalysis! */
  EigenAnalysisType eigenAnalysis(N);
  unsigned int      returncode = 0;
  returncode = eigenAnalysis.ComputeEigenValuesAndVectors(this->m_C, this->m_D, this->m_B);
  if (returncode != 0)
  {
    itkExceptionMacro(<< "EigenAnalysis failed while computing eigenvalue nr: " << returncode);
  }

  /** itk eigen analysis returns eigen vectors in rows... */
  this->m_B.inplace_transpose();

  /**  limit condition of C to 1e10 + 1, and avoid negative eigenvalues */
  const double largeNumber = 1e10;
  double       dmax = this->m_D.diagonal().max_value();
  double       dmin = this->m_D.diagonal().min_value();
  if (dmin < 0.0)
  {
    const double diagadd = dmax / largeNumber;
    for (unsigned int i = 0; i < N; ++i)
    {
      if (this->m_D[i] < 0.0)
      {
        this->m_D[i] = 0.0;
      }
      this->m_C[i][i] += diagadd;
      this->m_D[i] += diagadd;
    }
  }

  dmax = this->m_D.diagonal().max_value();
  dmin = this->m_D.diagonal().min_value();
  if (dmax > dmin * largeNumber)
  {
    const double diagadd = dmax / largeNumber - dmin;
    for (unsigned int i = 0; i < N; ++i)
    {
      this->m_C[i][i] += diagadd;
      this->m_D[i] += diagadd;
    }
  }

  /** the D matrix is supposed to contain the square root of the eigen values */
  for (unsigned int i = 0; i < N; ++i)
  {
    this->m_D[i] = std::sqrt(this->m_D[i]);
  }

  /** Keep for the user */
  this->m_CurrentMaximumD = this->m_D.diagonal().max_value();
  this->m_CurrentMinimumD = this->m_D.diagonal().min_value();

} // end UpdateBD


/**
 * **************** FixNumericalErrors ********************
 */

void
CMAEvolutionStrategyOptimizer::FixNumericalErrors(void)
{
  itkDebugMacro("FixNumericalErrors");

  /** Get the number of parameters from the cost function */
  const unsigned int numberOfParameters = this->GetScaledCostFunction()->GetNumberOfParameters();

  /** Some casts/aliases: */
  const unsigned int N = numberOfParameters;
  const double       c_sigma = this->m_ConjugateEvolutionPathConstant;
  const double       c_cov = this->m_CovarianceMatrixAdaptationConstant;
  const double       d_sigma = this->m_SigmaDampingConstant;
  const double       strange_factor = std::exp(0.05 + c_sigma / d_sigma);
  const double       strange_factor2 = std::exp(0.2 + c_sigma / d_sigma);
  const unsigned int nextit = this->m_CurrentIteration + 1;

  /** Check if m_MaximumDeviation and m_MinimumDeviation are satisfied. This
   * check is different depending on the m_UseCovarianceMatrixAdaptation flag */
  if (this->GetUseCovarianceMatrixAdaptation())
  {
    /** Check for too large deviation */
    for (unsigned int i = 0; i < N; ++i)
    {
      const double sqrtCii = std::sqrt(this->m_C[i][i]);
      const double actualDev = this->m_CurrentSigma * sqrtCii;
      if (actualDev > this->m_MaximumDeviation)
      {
        this->m_CurrentSigma = this->m_MaximumDeviation / sqrtCii;
      }
    }

    /** Check for too small deviation */
    bool minDevViolated = false;
    for (unsigned int i = 0; i < N; ++i)
    {
      const double sqrtCii = std::sqrt(this->m_C[i][i]);
      const double actualDev = this->m_CurrentSigma * sqrtCii;
      if (actualDev < this->m_MinimumDeviation)
      {
        this->m_CurrentSigma = this->m_MinimumDeviation / sqrtCii;
        minDevViolated = true;
      }
    }
    if (minDevViolated)
    {
      /** \todo: does this make sense if m_UseDecayingSigma == true??
       * Anyway, we have to do something, in order to satisfy the minimum deviation */
      this->m_CurrentSigma *= strange_factor;
    }
  }
  else
  {
    /** If no covariance matrix adaptation is used, the check becomes simpler */

    /** Check for too large deviation */
    double actualDev = this->m_CurrentSigma;
    if (actualDev > this->m_MaximumDeviation)
    {
      this->m_CurrentSigma = this->m_MaximumDeviation;
    }
    /** Check for too small deviation */
    bool minDevViolated = false;
    actualDev = this->m_CurrentSigma;
    if (actualDev < this->m_MinimumDeviation)
    {
      this->m_CurrentSigma = this->m_MinimumDeviation;
      minDevViolated = true;
    }
    if (minDevViolated)
    {
      /** \todo: does this make sense if m_UseDecayingSigma == true??
       * Anyway, we have to do something, in order to satisfy the minimum deviation */
      this->m_CurrentSigma *= strange_factor;
    }

  } // end else: no covariance matrix adaptation

  /** Adjust too low coordinate axis deviations that would cause numerical
   * problems (because of finite precision of the datatypes). This check
   * is different depending on the m_UseCovarianceMatrixAdaptation flag */
  const ParametersType & param = this->GetScaledCurrentPosition();
  bool                   numericalProblemsEncountered = false;
  if (this->GetUseCovarianceMatrixAdaptation())
  {
    /** Check for numerically too small deviation */
    for (unsigned int i = 0; i < N; ++i)
    {
      const double actualDev = 0.2 * this->m_CurrentSigma * std::sqrt(this->m_C[i][i]);
      if (param[i] == (param[i] + actualDev))
      {
        /** The parameters wouldn't change after perturbation, because
         * of too low precision. Increase the problematic diagonal element of C */
        this->m_C[i][i] *= (1.0 + c_cov);
        numericalProblemsEncountered = true;
      }
    } // end for i
  }
  else
  {
    const double actualDev = 0.2 * this->m_CurrentSigma;
    for (unsigned int i = 0; i < N; ++i)
    {
      if (param[i] == (param[i] + actualDev))
      {
        /** The parameters wouldn't change after perturbation, because
         * of too low precision. Increase the sigma (equivalent to
         * increasing a diagonal element of C^0.5).  */
        this->m_CurrentSigma *= std::sqrt(1.0 + c_cov);
        numericalProblemsEncountered = true;
      }
    }
  } // end else: no covariance matrix adaptation
  if (numericalProblemsEncountered)
  {
    /** \todo: does this make sense if m_UseDecayingSigma == true??
     * Anyway, we have to do something, in order to solve the numerical problems */
    this->m_CurrentSigma *= strange_factor;
  }

  /** Check if "main axis standard deviation sigma*D(i,i) has effect" (?),
   * with i = 1+floor(mod(countiter,N))
   * matlabcode: if all( xmean == xmean + 0.1*sigma*B*D(:,i) )
   * B*D(:,i) = i'th column of B times eigenvalue = i'th eigenvector * eigenvalue[i]
   * In the code below: colnr=i-1 (zero-based indexing). */
  bool               numericalProblemsEncountered2 = false;
  const unsigned int colnr = static_cast<unsigned int>(nextit % N);
  if (this->GetUseCovarianceMatrixAdaptation())
  {
    const double sigDcol = 0.1 * this->m_CurrentSigma * this->m_D[colnr];
    // const ParametersType actualDevVector = sigDcol * this->m_B.get_column(colnr);
    const ParametersType::VnlVectorType actualDevVector = sigDcol * this->m_B.get_column(colnr);
    if (param == (param + actualDevVector))
    {
      numericalProblemsEncountered2 = true;
    }
  }
  else
  {
    /** B and D are not used, so can be considered identity matrices.
     * This simplifies the check */
    const double sigDcol = 0.1 * this->m_CurrentSigma;
    if (param[colnr] == (param[colnr] + sigDcol))
    {
      numericalProblemsEncountered2 = true;
    }
  } // end else: no covariance matrix adaptation
  if (numericalProblemsEncountered2)
  {
    /** \todo: does this make sense if m_UseDecayingSigma == true??
     * Anyway, we have to do something, in order to solve the numerical problems */
    this->m_CurrentSigma *= strange_factor2;
  }

  /** Adjust step size in case of equal function values (flat fitness) */

  /** The indices of the two population members whose cost function will
   * be compared */
  const unsigned int populationMemberA = 0;
  const unsigned int populationMemberB =
    static_cast<unsigned int>(std::ceil(0.1 + static_cast<double>(this->m_PopulationSize) / 4.0));
  /** If they are the same: increase sigma with a magic factor */
  if (this->m_CostFunctionValues[populationMemberA].first == this->m_CostFunctionValues[populationMemberB].first)
  {
    this->m_CurrentSigma *= strange_factor2;
  }

  /** Check if the best function value changes over iterations */
  if (this->m_MeasureHistory.size() > 1)
  {
    const MeasureType maxhist = *max_element(this->m_MeasureHistory.begin(), this->m_MeasureHistory.end());
    const MeasureType minhist = *min_element(this->m_MeasureHistory.begin(), this->m_MeasureHistory.end());
    if (maxhist == minhist)
    {
      this->m_CurrentSigma *= strange_factor2;
    }
  }

} // end FixNumericalErrors


/**
 * ********************* TestConvergence ************************
 */

bool
CMAEvolutionStrategyOptimizer::TestConvergence(bool firstCheck)
{
  itkDebugMacro("TestConvergence");

  /** Get the number of parameters from the cost function */
  const unsigned int numberOfParameters = this->GetScaledCostFunction()->GetNumberOfParameters();

  /** Some casts/aliases: */
  const unsigned int N = numberOfParameters;

  /** Check if the maximum number of iterations will not be exceeded in the following iteration */
  if ((this->GetCurrentIteration() + 1) >= this->GetMaximumNumberOfIterations())
  {
    this->m_StopCondition = MaximumNumberOfIterations;
    return true;
  }

  /** Check if the step was not too large:
   * if ( sigma * sqrt(C[i,i]) > PositionToleranceMax*sigma0   for any i ) */
  const double tolxmax = this->m_PositionToleranceMax * this->m_InitialSigma;
  bool         stepTooLarge = false;
  if (this->GetUseCovarianceMatrixAdaptation())
  {
    for (unsigned int i = 0; i < N; ++i)
    {
      const double sqrtCii = std::sqrt(this->m_C[i][i]);
      const double stepsize = this->m_CurrentSigma * sqrtCii;
      if (stepsize > tolxmax)
      {
        stepTooLarge = true;
        break;
      }
    } // end for i
  }
  else
  {
    const double sqrtCii = 1.0;
    const double stepsize = this->m_CurrentSigma * sqrtCii;
    if (stepsize > tolxmax)
    {
      stepTooLarge = true;
    }
  } // end else: if no covariance matrix adaptation
  if (stepTooLarge)
  {
    this->m_StopCondition = PositionToleranceMax;
    return true;
  }

  /** Check for zero steplength (should never happen):
   * if ( sigma * D[i] <= 0  for all i  ) */
  bool zeroStep = false;
  if (this->GetUseCovarianceMatrixAdaptation())
  {
    if ((this->m_CurrentSigma * this->m_D.diagonal().max_value()) <= 0.0)
    {
      zeroStep = true;
    }
  }
  else
  {
    if (this->m_CurrentSigma <= 0.0)
    {
      zeroStep = true;
    }
  }
  if (zeroStep)
  {
    this->m_StopCondition = ZeroStepLength;
    return true;
  }

  /** The very first convergence check can not test for everything yet */
  if (firstCheck)
  {
    return false;
  }

  /** Check if the step was not too small:
   * if ( sigma * max( abs(p_c[i]), sqrt(C[i,i]) ) < PositionToleranceMin*sigma0  for all i ) */
  const double tolxmin = this->m_PositionToleranceMin * this->m_InitialSigma;
  bool         stepTooSmall = true;
  for (unsigned int i = 0; i < N; ++i)
  {
    const double pci = std::abs(this->m_EvolutionPath[i]);
    double       sqrtCii = 1.0;
    if (this->m_UseCovarianceMatrixAdaptation)
    {
      sqrtCii = std::sqrt(this->m_C[i][i]);
    }
    const double stepsize = this->m_CurrentSigma * std::max(pci, sqrtCii);
    if (stepsize > tolxmin)
    {
      stepTooSmall = false;
      break;
    }
  }
  if (stepTooSmall)
  {
    this->m_StopCondition = PositionToleranceMin;
    return true;
  }

  /** Check if the best function value changes over iterations */
  if (this->m_MeasureHistory.size() > 10)
  {
    const MeasureType maxhist = *max_element(this->m_MeasureHistory.begin(), this->m_MeasureHistory.end());
    const MeasureType minhist = *min_element(this->m_MeasureHistory.begin(), this->m_MeasureHistory.end());
    if ((maxhist - minhist) < this->m_ValueTolerance)
    {
      this->m_StopCondition = ValueTolerance;
      return true;
    }
  }

  return false;

} // end TestConvergence


} // end namespace itk
